2 Answers
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No, Lego does not sell custom printed bricks in small quantities (I suspect if you wanted thousands of the same design it would be a different story).

To get your search to work you need to use the advanced features, specifically by removing results that contain 3D: print on ABS plastic -3D

That turns up mostly links on screen printing, which would be awfully fiddly to do with Lego pieces.

A friend has a printer that will print properly on Lego, but it was not cheap to buy. IIRC it's an inkjet that takes special cartridges, presumably using the same sort of ink that the screen printers use. If you're willing to spend a couple of thousand dollars on buying a printer I can ask him for details. My solution would be to pay him to print bricks for me, as I also prefer not to use stickers, but I rarely use decorated bricks so that hasn't yet been necessary.

There are many LEGO customizers who do custom printing, however please note that all printing makes sense only if hundreds of the same design are requested. Otherwise there is quite a bit of work involved with designing, formatting, color-matching and aligning each print. However if you have the money and willing to pay a few hundred bucks set-up fee for each design, there are options.

Please note that most LEGO customizers won't be willing to do exact replicas of LEGO's own design, as it is illegal to reproduce and sell them.

You might be better off getting water-slide decals, and even designing them yourself. The end result looks almost the same as direct printing on elements, although they will be more delicate.

If you are interested to get your own printer for LEGO pieces they are called pad-printing machines. They are expensive, but prices are coming down, especially on used ones.